Is despairing of attempts to mend broken family ties—due to family disputes and verbal abuse—considered severing kinship ties? And what is the best solution to this problem?

The aunt resorting to the judiciary to claim her right is not wrong. Your duty was to reconcile her and her brother amicably and to support the one who has the right. If it is not clear to you who is in the right, then leave the matter to the judiciary. We advise you to seek the mediation of good-doers to reconcile all of you, resolve the dispute, and respond to offense with forgiveness and reconciliation, for "whoever pardons and makes reconciliation - his reward is due from Allah." If the aunt accepts, then all praise is due to Allah; otherwise, you have fulfilled your obligation.